Francesca Balestrieri: Uniform bounds and effectivity results for singular K3 surfaces

Abstract

This is joint work with Alexis Johnson and Rachel Newton. Let \(k\) be a number field. We give an explicit bound, depending only on \([k : Q]\) and the discriminant of the Néron–Severi lattice, on the size of the Brauer group of a K3 surface \(X/k\) that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer–Manin set for such a variety is effectively computable. In addition, we show how to obtain a bound, depending only on \([k : Q]\), on the number of C-isomorphism classes of singular K3 surfaces defined over k, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.